cp's OEIS Frontend

A376102 Array read by ascending antidiagonals: A(n,k) = k*2^(n+1) + 1.

%I A376102 #38 Jul 18 2025 09:33:52
%S A376102 1,1,3,1,5,5,1,9,9,7,1,17,17,13,9,1,33,33,25,17,11,1,65,65,49,33,21,
%T A376102 13,1,129,129,97,65,41,25,15,1,257,257,193,129,81,49,29,17,1,513,513,
%U A376102 385,257,161,97,57,33,19,1,1025,1025,769,513,321,193,113,65,37,21
%N A376102 Array read by ascending antidiagonals: A(n,k) = k*2^(n+1) + 1.
%C A376102 In 1747, Euler showed that any factor of a Fermat number A000215(n) is of the form k*2^(n+1) + 1. See Wells at p. 148.
%D A376102 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 70-71, 237-242.
%D A376102 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 136.
%D A376102 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987.
%F A376102 G.f.: (1 - 2*x + y)/((1 - x)*(1 - 2*x)*(1 - y)^2).
%F A376102 E.g.f.: exp(x+y)*(1 + 2*exp(x)*y).
%F A376102 Sum_{0<=k<=n} A(n-k,k) = A000295(n+2).
%F A376102 A(n,1) = A000051(n+1).
%F A376102 A(n,3) = A004119(n+2).
%F A376102 A(n,n) = A000337(n+1).
%e A376102 The array begins as:
%e A376102   1,   3,   5,   7,   9,  11,  13, ...
%e A376102   1,   5,   9,  13,  17,  21,  25, ...
%e A376102   1,   9,  17,  25,  33,  41,  49, ...
%e A376102   1,  17,  33,  49,  65,  81,  97, ...
%e A376102   1,  33,  65,  97, 129, 161, 193, ...
%e A376102   1,  65, 129, 193, 257, 321, 385, ...
%e A376102   1, 129, 257, 385, 513, 641, 769, ...
%e A376102   ...
%t A376102 A[n_,k_]:=k*2^(n+1)+1; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten
%Y A376102 Cf. A000079, A000215, A000295.
%Y A376102 Cf. A000012 (k=0), A000051, A000337, A004119, A005408 (n=0), A016813 (n=1), A017077 (n=2), A158057 (n=3).
%K A376102 nonn,easy,tabl
%O A376102 0,3
%A A376102 _Stefano Spezia_, Sep 14 2024